ODE
\[ (y(x)+x)^2 y'(x)^2=y(x)^2 \] ODE Classification
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
Book solution method
No Missing Variables ODE, Solve for \(y'\)
Mathematica ✓
cpu = 0.0423201 (sec), leaf count = 64
\[\left \{\left \{y(x)\to -\sqrt {e^{2 c_1}+x^2}-x\right \},\left \{y(x)\to \sqrt {e^{2 c_1}+x^2}-x\right \},\left \{y(x)\to \frac {x}{W\left (e^{-c_1} x\right )}\right \}\right \}\]
Maple ✓
cpu = 0.189 (sec), leaf count = 29
\[ \left \{ x- \left ( \ln \left ( y \left ( x \right ) \right ) +{\it \_C1} \right ) y \left ( x \right ) =0,x+{\frac {y \left ( x \right ) }{2}}-{\frac {{\it \_C1}}{y \left ( x \right ) }}=0 \right \} \] Mathematica raw input
DSolve[(x + y[x])^2*y'[x]^2 == y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> -x - Sqrt[E^(2*C[1]) + x^2]}, {y[x] -> -x + Sqrt[E^(2*C[1]) + x^2]}, {y[x] -> x/ProductLog[x/E^C[1]]}}
Maple raw input
dsolve((x+y(x))^2*diff(y(x),x)^2 = y(x)^2, y(x),'implicit')
Maple raw output
x-(ln(y(x))+_C1)*y(x) = 0, x+1/2*y(x)-1/y(x)*_C1 = 0